Is it possible for a single object to have more than one mesh?
No, you can't give one object two mesh datablocks at once. 

However, if you need to do something like packing UVs of two meshes together you can assign a vertex group to all the vertices in the active mesh. Then you should join the meshes (CtrlJ), do the unwrap or whatever you needed to do, deselect everything, select the vertices in the group, and separate them (P > Selection). That way you have your original object back, but unwrapped together. The Texture Atlas addon can also do this.

If you need to move two meshes together, parent them both to an Empty ( CtrlP).